Selecta Mathematica

, 13:369

Gelfand transforms and Crofton formulas

Article

DOI: 10.1007/s00029-007-0045-5

Cite this article as:
Paiva, J.C.Á. & Fernandes, E. Sel. math., New ser. (2008) 13: 369. doi:10.1007/s00029-007-0045-5

Abstract.

The term integral geometry has come to describe two different fields of research: one, geometrical, based on the works of Blaschke, Chern, and Santaló, and another, analytical, based on the works of Radon, John, Helgason, and Gelfand. In this paper we bridge the gap by showing that classical integral-geometric formulas such as those of Crofton, Cauchy, and Chern can be easily and systematically obtained through the study of Radon-type transforms on double fibrations. The methods also allow us to extend these formulas to non-homogeneous settings where group-theoretic techniques are no longer useful. To illustrate this point, we construct all Finsler metrics on projective space such that hyperplanes are area-minimizing and extend the theory of Crofton densities developed by Busemann, Pogorelov, Gelfand, and Smirnov.

Pensar es olvidar diferencias. . .

Jorge Luis Borges

Mathematics Subject Classification (2000).

Primary 53C65Secondary 53C60

Keywords.

Integral geometryCrofton formulasdouble fibrationsGelfand transformsFinsler manifoldHolmes–Thompson area

Copyright information

© Birkhaueser 2008

Authors and Affiliations

  1. 1.Laboratoire Paul PainlevéUniversité des Sciences et Technologies de LilleVilleneuve d’Ascq CedexFrance
  2. 2.MULTITEL Research Centre, Parc InitialisMonsBelgium